Ilya Gekhtman, Linearly growing injectivity radius in negatively curved manifolds with small скачать в хорошем качестве

Ilya Gekhtman, Linearly growing injectivity radius in negatively curved manifolds with small

Let X be a proper geodesic Gromov hyperbolic space whose isometry group contains a uniform lattice Γ. For instance, X could be a negatively curved contractible manifold or a Cayley graph of a hyperbolic group. Let H be a discrete subgroup of isometries of X with critical exponent (exponential growth rate) strictly less than half of the growth rate of Γ. We show that the injectivity radius of X/H grows linearly along almost every geodesic in X (with respect to the Patterson-Sullivan measure on the Gromov boundary of X). The proof will involve an elementary analysis of a novel concept called the “sublinearly horosherical limit set” of H which is a generalization of the classical concept of “horospherical limit set” for Kleinian groups.